Problem: You have found the following ages (in years) of all 6 turtles at your local zoo: $ 76,\enspace 35,\enspace 38,\enspace 71,\enspace 86,\enspace 66$ What is the average age of the turtles at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Because we have data for all 6 turtles at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{76 + 35 + 38 + 71 + 86 + 66}{{6}} = {62\text{ years old}} $ Find the squared deviations from the mean for each turtle. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $76$ years $14$ years $196$ years $^2$ $35$ years $-27$ years $729$ years $^2$ $38$ years $-24$ years $576$ years $^2$ $71$ years $9$ years $81$ years $^2$ $86$ years $24$ years $576$ years $^2$ $66$ years $4$ years $16$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{196} + {729} + {576} + {81} + {576} + {16}} {{6}} $ $ {\sigma^2} = \dfrac{{2174}}{{6}} = {362.33\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{362.33\text{ years}^2}} = {19\text{ years}} $ The average turtle at the zoo is 62 years old. There is a standard deviation of 19 years.